Properties

Label 1728.3.h.g
Level $1728$
Weight $3$
Character orbit 1728.h
Analytic conductor $47.085$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{5} + \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{5} + \beta_{3} q^{7} - \beta_{7} q^{11} + 3 \beta_{2} q^{13} - 3 \beta_{6} q^{17} + 5 \beta_1 q^{19} + 5 \beta_{5} q^{23} - q^{25} + 8 \beta_{4} q^{29} + 8 \beta_{3} q^{31} - \beta_{7} q^{35} + 19 \beta_{2} q^{37} + 2 \beta_{6} q^{41} + 34 \beta_1 q^{43} - \beta_{5} q^{47} - 46 q^{49} - 2 \beta_{4} q^{53} + 24 \beta_{3} q^{55} - 9 \beta_{7} q^{59} + 13 \beta_{2} q^{61} - 3 \beta_{6} q^{65} + 19 \beta_1 q^{67} - 14 \beta_{5} q^{71} - 25 q^{73} - 3 \beta_{4} q^{77} + 81 \beta_{3} q^{79} + 12 \beta_{7} q^{83} + 72 \beta_{2} q^{85} - \beta_{6} q^{89} + 9 \beta_1 q^{91} + 5 \beta_{5} q^{95} - 71 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{25} - 368 q^{49} - 200 q^{73} - 568 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{24}^{4} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{24}^{6} + 2\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -4\zeta_{24}^{7} + 2\zeta_{24}^{5} + 2\zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -4\zeta_{24}^{7} - 2\zeta_{24}^{5} + 2\zeta_{24}^{3} - 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 6\zeta_{24}^{5} + 6\zeta_{24}^{3} - 6\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -6\zeta_{24}^{5} + 6\zeta_{24}^{3} + 6\zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{7} - \beta_{6} - 3\beta_{5} + 3\beta_{4} ) / 24 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} ) / 12 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} + \beta_{6} - 3\beta_{5} + 3\beta_{4} ) / 24 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} + \beta_{6} - 3\beta_{5} - 3\beta_{4} ) / 24 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
0.258819 + 0.965926i
0.258819 0.965926i
0.965926 0.258819i
0.965926 + 0.258819i
−0.258819 0.965926i
−0.258819 + 0.965926i
−0.965926 + 0.258819i
−0.965926 0.258819i
0 0 0 −4.89898 0 −1.73205 0 0 0
161.2 0 0 0 −4.89898 0 −1.73205 0 0 0
161.3 0 0 0 −4.89898 0 1.73205 0 0 0
161.4 0 0 0 −4.89898 0 1.73205 0 0 0
161.5 0 0 0 4.89898 0 −1.73205 0 0 0
161.6 0 0 0 4.89898 0 −1.73205 0 0 0
161.7 0 0 0 4.89898 0 1.73205 0 0 0
161.8 0 0 0 4.89898 0 1.73205 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.3.h.g 8
3.b odd 2 1 inner 1728.3.h.g 8
4.b odd 2 1 inner 1728.3.h.g 8
8.b even 2 1 inner 1728.3.h.g 8
8.d odd 2 1 inner 1728.3.h.g 8
12.b even 2 1 inner 1728.3.h.g 8
24.f even 2 1 inner 1728.3.h.g 8
24.h odd 2 1 inner 1728.3.h.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.3.h.g 8 1.a even 1 1 trivial
1728.3.h.g 8 3.b odd 2 1 inner
1728.3.h.g 8 4.b odd 2 1 inner
1728.3.h.g 8 8.b even 2 1 inner
1728.3.h.g 8 8.d odd 2 1 inner
1728.3.h.g 8 12.b even 2 1 inner
1728.3.h.g 8 24.f even 2 1 inner
1728.3.h.g 8 24.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{2} - 24 \) Copy content Toggle raw display
\( T_{7}^{2} - 3 \) Copy content Toggle raw display
\( T_{11}^{2} - 72 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} - 24)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 3)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 72)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 27)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 648)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 25)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 600)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 1536)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 192)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1083)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 288)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 1156)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 24)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 96)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 5832)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 507)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 361)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 4704)^{4} \) Copy content Toggle raw display
$73$ \( (T + 25)^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} - 19683)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 10368)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 72)^{4} \) Copy content Toggle raw display
$97$ \( (T + 71)^{8} \) Copy content Toggle raw display
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